The Pentagram of Venus

4 January, 2014


This image, made by Greg Egan, shows the orbit of Venus.

Look down on the plane of the Solar System from above the Earth. Track the Earth so it always appears directly below you, but don’t turn along with it. With the passage of each year, you will see the Sun go around the Earth. As the Sun goes around the Earth 8 times, Venus goes around the Sun 13 times, and traces out the pretty curve shown here.

It’s called the pentagram of Venus, because it has 5 ‘lobes’ where Venus makes its closest approach to Earth. At each closest approach, Venus move backwards compared to its usual motion across the sky: this is called retrograde motion.

Actually, what I just said is only approximately true. The Earth orbits the Sun once every


days. Venus orbits the Sun once every


days. So, Venus orbits the Sun in

224.701 / 365.256 ≈ 0.615187

Earth years. And here’s the cool coincidence:

8/13 ≈ 0.615385

That’s pretty close! So in 8 Earth years, Venus goes around the Sun almost 13 times. Actually, it goes around 13.004 times.

During this 8-year cycle, Venus gets as close as possible to the Earth about

13 – 8 = 5

times. And each time it does, Venus moves to a new lobe of the pentagram of Venus! This new lobe is

8 – 5 = 3

steps ahead of the last one. Check to make sure:

That’s why they call it the pentagram of Venus!

When Venus gets as close as possible to us, we see it directly in front of the Sun. This is called an inferior conjunction. Astronomers have names for all of these things:

So, every 8 years there are about 5 inferior conjunctions of Venus.

Puzzle 1: Suppose the Earth orbits the Sun n times while another planet, closer to the Sun, orbits it m times. Under what conditions does the ‘generalized pentagram’ have k = mn lobes? (The pentagram of Venus has 5 = 13 – 8 lobes.)

Puzzle 2: Under what conditions does the planet move forward j = nk steps each time it reaches a new lobe? (Venus moves ahead 3 = 8 – 5 steps each time.)

Now, I’m sure you’ve noticed that these numbers:

3, 5, 8, 13

are consecutive Fibonacci numbers.

Puzzle 3: Is this just a coincidence?

As you may have heard, ratios of consecutive Fibonacci numbers give the best approximations to the golden ratio φ = (√5 – 1)/2. This number actually plays a role in celestial mechanics: the Kolmogorov–Arnol’d–Moser theorem says two systems vibrating with frequencies having a ratio equal to φ are especially stable against disruption by resonances, because this number is hard to approximate well by rationals. But the Venus/Earth period ratio 0.615187 is actually closer to the rational number 8/13 ≈ 0.615385 than φ ≈ 0.618034. So if this period ratio is trying to avoid rational numbers by being equal to φ, it’s not doing a great job!

It’s all rather tricky, because sometimes rational numbers cause destabilizing resonances, as we see in the gaps of Saturn’s rings:

whereas other times rational numbers stabilize orbits, as with the moons of Jupiter:

I’ve never understood this, and I’m afraid no amount of words will help me: I’ll need to dig into the math.

Given my fascination with rolling circles and the number 5, I can’t believe that I learned about the pentagram of Venus only recently! It’s been known at least for centuries, perhaps millennia. Here’s a figure from James Ferguson’s 1799 book Astronomy Explained Upon Sir Isaac Newton’s Principles:

Naturally, some people get too excited about all this stuff—the combination of Venus, Fibonacci numbers, the golden ratio, and a ‘pentagram’ overloads their tiny brains. Some claim the pentagram got its origin from this astronomical phenomenon. I doubt we’ll ever know. Some get excited about the fact that a Latin name for the planet Venus is Lucifer. Lucifer, pentagrams… get it?

I got the above picture from here:

Venus and the pentagram, Grand Lodge of British Columbia and Yukon.

This website is defending the Freemasons against accusations of Satanism!

On a sweeter note, the pentagram of Venus is also called the rose of Venus. You can buy a pendant in this pattern:

It’s pretty—but according to the advertisement, that’s not all! It’s also “an energetic tool that creates a harmonising field of Negative Ion around our body to support and balance our own magnetic field and aura.”

In The Da Vinci Code, someone claims that Venus traces “a perfect pentacle across the ecliptic sky every 8 years.”

But it’s not perfect! Every 8 years, Venus goes around the Sun 13.004 times. So the whole pattern keeps shifting. It makes a full turn about once every 160 years. You can see this slippage using this nice applet, especially if you crank up the speed:

• Steven Deutch, The (almost) Venus-Earth pentagram.

Also, the orbits of Earth and Venus aren’t perfect circles!

But still, it’s fun. The universe is full of mathematical beauty. It seems we need to get closer and closer to the fundamental laws of nature to make the math and the universe match more and more accurately. Maybe that’s what ‘fundamental laws’ means. But the universe is also richly packed with beautiful approximate mathematical patterns, stacked on top of each other in a dizzying way.


Talk at the SETI Institute

5 December, 2013

SETI means ‘Search for Extraterrestrial Intelligence’. I’m giving a talk at the SETI Institute on Tuesday December 17th, from noon to 1 pm. You can watch it live, watch it later on their YouTube channel, or actually go there and see it. It’s free, and you can just walk in at 189 San Bernardo Avenue in Mountain View, California, but please register if you can.

Life’s Struggle to Survive

When pondering the number of extraterrestrial civilizations, it is worth noting that even after it got started, the success of life on Earth was not a foregone conclusion. We recount some thrilling episodes from the history of our planet, some well-documented but others merely theorized: our collision with the planet Theia, the oxygen catastrophe, the snowball Earth events, the Permian-Triassic mass extinction event, the asteroid that hit Chicxulub, and more, including the global warming episode we are causing now. All of these hold lessons for what may happen on other planets.

If you know interesting things about these or other ‘close calls’, please tell me! I’m still preparing my talk, and there’s room for more fun facts. I’ll make my slides available when they’re ready.

The SETI Institute looks like an interesting place, and my host, Adrian Brown, is an expert on the poles of Mars. I’ve been fascinated about the water there, and I’ll definitely ask him about this paper:

• Adrian J. Brown, Shane Byrne, Livio L. Tornabene and Ted Roush, Louth crater: Evolution of a layered water ice mound, Icarus 196 (2008), 433–445.

Louth Crater is a fascinating place. Here’s a photo:

By the way, I’ll be in Berkeley from December 14th to 21st, except for a day trip down to Mountain View for this talk. I’ll be at the Machine Intelligence Research Institute talking to Eliezer Yudkowsky, Paul Christiano and others at a Workshop on Probability, Logic and Reflection. This invitation arose from my blog post here:

Probability theory and the undefinability of truth.

If you’re in Berkeley and you want to talk, drop me a line. I may be too busy, but I may not.

The Search For Budget-Conscious Life

18 May, 2013


Lisa and I had dinner with Gregory Benford and his wife when I visited U.C. Irvine a couple of weekends ago, and he raised an interesting point. So far, radio searches for extraterrestrial life have only seen puzzling brief signals – not long transmissions. But what if this is precisely what we should expect?

A provocative example is Sullivan, et al. (1997). This survey lasted about 2.5 hours, with 190 1.2 minute integrations. With many repeat observations, they saw nothing that did not seem manmade. However, they “recorded intriguing, non-repeatable, narrowband signals, apparently not of manmade origin and with some degree of concentration toward the galactic plane…” Similar searches also saw one-time signals, not repeated (Shostak & Tarter, 1985; Gray & Marvel, 2001 Gray, 2001). These searches had slow times to revisit or reconfirm, often days (Tarter, 2001). Overall, few searches lasted more than hour, with lagging confirmation checks (Horowitz & Sagan, 1993). Another striking example is the “WOW” signal seen at the Ohio SETI site…

That’s a quote from a paper Benford wrote with his brother and nephew:

• Gregory Benford, James Benford, and Dominic Benford, Searching for cost optimized interstellar beacons.

They claim the cheapest way a civilization could communicate to lots of planets is a pulsed, broadband, narrowly focused microwave beam that scans the sky. So, for anyone receiving this signal, there would be a lot of time between pulses. That might explain some of the above mysteries, or this one:

As an example of using cost optimized beacon analysis for SETI purposes, consider in detail the puzzling transient bursting radio source, GCRT J17445-3009, which has extremely unusual properties. It was discovered in 2002 in the direction of the Galactic Center (1.25° south of GC) at 330 MHz in a VLA observation and subsequently re-observed in 2003 and 2004 in GMRT observations (Hyman, 2005, 2006, 2007). It is a pulsed coherent source, with the ‘burst’ lasting as much as 10 minutes, with 77-minute period. Averaged over all observations, Hyman et al. give a duty cycle of 7% (1/14), although since some observations may have missed part of bursts, the duty cycle might be as high as 13%.

Even if these are red herrings, it seems very smart to figure out the cheapest ways to transmit signals and use that to guess what signals we should look for. We can easily make the mistake of assuming all extraterrestrial civilizations who bother to send signals through space will be willing to beam signals of enormous power toward us all the time. That could be true of some, but not necessarily all.

The cost analysis is here:

• James Benford, Gregory Benford, Dominic Benford, Messaging with cost optimized interstellar beacons.

and you can see a summary in this talk by Gregory’s brother James, who works on high-power microwave technologies:

The Planck Mission

22 March, 2013

Yesterday, the Planck Mission released a new map of the cosmic microwave background radiation:

380,000 years after the Big Bang, the Universe cooled down enough for protons and electrons to settle down and combine into hydrogen atoms. Protons and electrons are charged, so back when they were freely zipping around, no light could go very far without getting absorbed and then re-radiated. When they combined into neutral hydrogen atoms, the Universe soon switched to being almost transparent… as it is today. So the light emitted from that time is still visible now!

And it would look like this picture here… if you could see microwaves.

When this light was first emitted, it would have looked white to our eyes, since the temperature of the Universe was about 4000 kelvin. That’s the temperature when half the hydrogen atoms split apart into electrons and protons. 4200 kelvin looks like a fluorescent light; 2800 kelvin like an incandescent bulb, rather yellow.

But as the Universe expanded, this light got stretched out to orange, red, infrared… and finally a dim microwave glow, invisible to human eyes. The average temperature of this glow is very close to absolute zero, but it’s been measured very precisely: 2.725 kelvin.

But the temperature of the glow is not the same in every direction! There are tiny fluctuations! You can see them in this picture. The colors here span a range of ± .0002 kelvin.

These fluctuations are very important, because they were later amplified by gravity, with denser patches of gas collapsing under their own gravitational attraction (thanks in part to dark matter), and becoming even denser… eventually leading to galaxies, stars and planets, you and me.

But where did these fluctuations come from? I suspect they started life as quantum fluctuations in an originally completely homogeneous Universe. Quantum mechanics takes quite a while to explain – but in this theory a situation can be completely symmetrical, yet when you measure it, you get an asymmetrical result. The universe is then a ‘sum’ of worlds where these different results are seen. The overall universe is still symmetrical, but each observer sees just a part: an asymmetrical part.

If you take this seriously, there are other worlds where fluctuations of the cosmic microwave background radiation take all possible patterns… and form galaxies in all possible patterns. So while the universe as we see it is asymmetrical, with galaxies and stars and planets and you and me arranged in a complicated and seemingly arbitrary way, the overall universe is still symmetrical – perfectly homogeneous!

That seems very nice to me. But the great thing is, we can learn more about this, not just by chatting, but by testing theories against ever more precise measurements. The Planck Mission is a great improvement over the Wilkinson Microwave Anisotropy Probe (WMAP), which in turn was a huge improvement over the Cosmic Background Explorer (COBE):

Here is some of what they’ve learned:

• It now seems the Universe is 13.82 ± 0.05 billion years old. This is a bit higher than the previous estimate of 13.77 ± 0.06 billion years, due to the Wilkinson Microwave Anisotropy Probe.

• It now seems the rate at which the universe is expanding, known as Hubble’s constant, is 67.15 ± 1.2 kilometers per second per megaparsec. A megaparsec is roughly 3 million light-years. This is less than earlier estimates using space telescopes, such as NASA’s Spitzer and Hubble.

• It now seems the fraction of mass-energy in the Universe in the form of dark matter is 26.8%, up from 24%. Dark energy is now estimated at 68.3%, down from 71.4%. And normal matter is now estimated at 4.9%, up from 4.6%.

These cosmological parameters, and a bunch more, are estimated here:

Planck 2013 results. XVI. Cosmological parameters.

It’s amazing how we’re getting more and more accurate numbers for these basic facts about our world! But the real surprises lie elsewhere…

A lopsided universe, with a cold spot?


The Planck Mission found two big surprises in the cosmic microwave background:

• This radiation is slightly different on opposite sides of the sky! This is not due to the fact that the Earth is moving relative to the average position of galaxies. That fact does make the radiation look hotter in the direction we’re moving. But that produces a simple pattern called a ‘dipole moment’ in the temperature map. If we subtract that out, it seems there are real differences between two sides of the Universe… and they are complex, interesting, and not explained by the usual theories!

• There is a cold spot that seems too big to be caused by chance. If this is for real, it’s the largest thing in the Universe.

Could these anomalies be due to experimental errors, or errors in data analysis? I don’t know! They were already seen by the Wilkinson Microwave Anisotropy Probe; for example, here is WMAP’s picture of the cold spot:

The Planck Mission seems to be seeing them more clearly with its better measurements. Paolo Natoli, from the University of Ferrara writes:

The Planck data call our attention to these anomalies, which are now more important than ever: with data of such quality, we can no longer neglect them as mere artefacts and we must search for an explanation. The anomalies indicate that something might be missing from our current understanding of the Universe. We need to find a model where these peculiar traits are no longer anomalies but features predicted by the model itself.

For a lot more detail, see this paper:

Planck 2013 results. XXIII. Isotropy and statistics of the CMB.

(I apologize for not listing the authors on these papers, but there are hundreds!) Let me paraphrase the abstract for people who want just a little more detail:

Many of these anomalies were previously observed in the Wilkinson Microwave Anisotropy Probe data, and are now confirmed at similar levels of significance (around 3 standard deviations). However, we find little evidence for non-Gaussianity with the exception of a few statistical signatures that seem to be associated with specific anomalies. In particular, we find that the quadrupole-octopole alignment is also connected to a low observed variance of the cosmic microwave background signal. The dipolar power asymmetry is now found to persist to much smaller angular scales, and can be described in the low-frequency regime by a phenomenological dipole modulation model. Finally, it is plausible that some of these features may be reflected in the angular power spectrum of the data which shows a deficit of power on the same scales. Indeed, when the power spectra of two hemispheres defined by a preferred direction are considered separately, one shows evidence for a deficit in power, whilst its opposite contains oscillations between odd and even modes that may be related to the parity violation and phase correlations also detected in the data. Whilst these analyses represent a step forward in building an understanding of the anomalies, a satisfactory explanation based on physically motivated models is still lacking.

If you’re a scientist, your mouth should be watering now… your tongue should be hanging out! If this stuff holds up, it’s amazing, because it would call for real new physics.

I’ve heard that the difference between hemispheres might fit the simplest homogeneous but not isotropic solutions of general relativity, the Bianchi models. However, this is something one should carefully test using statistics… and I’m sure people will start doing this now.

As for the cold spot, the best explanation I can imagine is some sort of mechanism for producing fluctuations very early on… so that these fluctuations would get blown up to enormous size during the inflationary epoch, roughly between 10-36 and 10-32 seconds after the Big Bang. I don’t know what this mechanism would be!

There are also ways of trying to ‘explain away’ the cold spot, but even these seem jaw-droppingly dramatic. For example, an almost empty region 150 megaparsecs (500 million light-years) across would tend to cool down cosmic microwave background radiation coming through it. But it would still be the largest thing in the Universe! And such an unusual void would seem to beg for an explanation of its own.

Particle physics

The Planck Mission also shed a lot of light on particle physics, and especially on inflation. But, it mainly seems to have confirmed what particle physicists already suspected! This makes them rather grumpy, because these days they’re always hoping for something new, and they’re not getting it.

We can see this at Jester’s blog Résonaances, which also gives a very nice, though technical, summary of what the Planck Mission did for particle physics:

From a particle physicist’s point of view the single most interesting observable from Planck is the notorious N_{\mathrm{eff}}. This observable measures the effective number of degrees of freedom with sub-eV mass that coexisted with the photons in the plasma at the time when the CMB was formed (see e.g. my older post for more explanations). The standard model predicts N_{\mathrm{eff}} \approx 3, corresponding to the 3 active neutrinos. Some models beyond the standard model featuring sterile neutrinos, dark photons, or axions could lead to N_{\mathrm{eff}} > 3, not necessarily an integer. For a long time various experimental groups have claimed N_{\mathrm{eff}} much larger than 3, but with an error too large to blow the trumpets. Planck was supposed to sweep the floor and it did. They find

N_{\mathrm{eff}} = 3 \pm 0.5,

that is, no hint of anything interesting going on. The gurgling sound you hear behind the wall is probably your colleague working on sterile neutrinos committing a ritual suicide.

Another number of interest for particle theorists is the sum of neutrino masses. Recall that oscillation experiments tell us only about the mass differences, whereas the absolute neutrino mass scale is still unknown. Neutrino masses larger than 0.1 eV would produce an observable imprint into the CMB. [....] Planck sees no hint of neutrino masses and puts the 95% CL limit at 0.23 eV.

Literally, the most valuable Planck result is the measurement of the spectral index n_s, as it may tip the scale for the Nobel committee to finally hand out the prize for inflation. Simplest models of inflation (e.g., a scalar field φ with a φn potential slowly changing its vacuum expectation value) predicts the spectrum of primordial density fluctuations that is adiabatic (the same in all components) and Gaussian (full information is contained in the 2-point correlation function). Much as previous CMB experiments, Planck does not see any departures from that hypothesis. A more quantitative prediction of simple inflationary models is that the primordial spectrum of fluctuations is almost but not exactly scale-invariant. More precisely, the spectrum is of the form

\displaystyle{ P \sim (k/k_0)^{n_s-1} }

with n_s close to but typically slightly smaller than 1, the size of n_s being dependent on how quickly (i.e. how slowly) the inflaton field rolls down its potential. The previous result from WMAP-9,

n_s=0.972 \pm 0.013

(n_s =0.9608 \pm 0.0080 after combining with other cosmological observables) was already a strong hint of a red-tilted spectrum. The Planck result

n_s = 0.9603 \pm 0.0073

(n_s =0.9608 \pm 0.0054 after combination) pushes the departure of n_s - 1 from zero past the magic 5 sigma significance. This number can of course also be fitted in more complicated models or in alternatives to inflation, but it is nevertheless a strong support for the most trivial version of inflation.


In summary, the cosmological results from Planck are really impressive. We’re looking into a pretty wide range of complex physical phenomena occurring billions of years ago. And, at the end of the day, we’re getting a perfect description with a fairly simple model. If this is not a moment to cry out “science works bitches”, nothing is. Particle physicists, however, can find little inspiration in the Planck results. For us, what Planck has observed is by no means an almost perfect universe… it’s rather the most boring universe.

I find it hilarious to hear someone complain that the universe is “boring” on a day when astrophysicists say they’ve discovered the universe is lopsided and has a huge cold region, the largest thing ever seen by humans!

However, particle physicists seem so far rather skeptical of these exciting developments. Is this sour grapes, or are they being wisely cautious?

Time, as usual, will tell.

Black Holes and the Golden Ratio

28 February, 2013


The golden ratio shows up in the physics of black holes!

Or does it?

Most things get hotter when you put more energy into them. But systems held together by gravity often work the other way. For example, when a red giant star runs out of fuel and collapses, its energy goes down but its temperature goes up! We say these systems have a negative specific heat.

The prime example of a system held together by gravity is a black hole. Hawking showed—using calculations, not experiments—that a black hole should not be perfectly black. It should emit ‘Hawking radiation’. So it should have a very slight glow, as if it had a temperature above zero. For a black hole the mass of the Sun this temperature would be just 6 × 10-8 kelvin.

This is absurdly chilly, much colder than the microwave background radiation left over from the Big Bang. So in practice, such a black hole will absorb stuff—stars, nearby gas and dust, starlight, microwave background radiation, and so on—and grow bigger. But if we could protect it from all this stuff, and put it in a very cold box, it would slowly shrink by emitting radiation and losing energy, and thus mass. As it lost energy, its temperature would go up. The less energy it has, the hotter it gets: a negative specific heat! Eventually, as it shrinks to nothing, it should explode in a very hot blast.

But for a spinning black hole, things are more complicated. If it spins fast enough, its specific heat will be positive, like a more ordinary object.

And according to a 1989 paper by Paul Davies, the transition to positive specific heat happens at a point governed by the golden ratio! He claimed that in units where the speed of light and gravitational constant are 1, it happens when

\displaystyle{  \frac{J^2}{M^4} = \frac{\sqrt{5} - 1}{2}  }

Here J is the black hole’s angular momentum, M is its mass, and

\displaystyle{ \frac{\sqrt{5} - 1}{2} = 0.6180339\dots }

is a version of the golden ratio! This is for black holes with no electric charge.

Unfortunately, this claim is false. Cesar Uliana, who just did a master’s thesis on black hole thermodynamics, pointed this out in the comments below after I posted this article.

And curiously, twelve years before writing this paper with the mistake in it, Davies wrote a paper that got the right answer to the same problem! It’s even mentioned in the abstract.

The correct constant is not the golden ratio! The correct constant is smaller:

\displaystyle{ 2 \sqrt{3} - 3 = 0.46410161513\dots }

However, Greg Egan figured out the nature of Davies’ slip, and thus discovered how the golden ratio really does show up in black hole physics… though in a more quirky and seemingly less significant way.

As usually defined, the specific heat of a rotating black hole measures the change in internal energy per change in temperature while angular momentum is held constant. But Davies looked at the change in internal energy per change in temperature while the ratio of angular momentum to mass is held constant. It’s this modified quantity that switches from positive to negative when

\displaystyle{  \frac{J^2}{M^4} = \frac{\sqrt{5} - 1}{2} }

In other words:

Suppose we gradually add mass and angular momentum to a black hole while not changing the ratio of angular momentum, J, to mass, M. Then J^2/M^4 gradually drops. As this happens, the black hole’s temperature increases until

\displaystyle{ \frac{J^2}{M^4} = \frac{\sqrt{5} - 1}{2} }

in units where the speed of light and gravitational constant are 1. And then it starts dropping!

What does this mean? It’s hard to tell. It doesn’t seem very important, because it seems there’s no good physical reason for the ratio of J to M to stay constant. In particular, as a black hole shrinks by emitting Hawking radiation, this ratio goes to zero. In other words, the black hole spins down faster than it loses mass.


Discussions of black holes and the golden ratio can be found in a variety of places. Mario Livio is the author of The Golden Ratio, and also an astrophysicist, so it makes sense that he would be interested in this connection. He wrote about it here:

• Mario Livio, The golden ratio and astronomy, Huffington Post, 22 August 2012.

Marcus Chown, the main writer on cosmology for New Scientist, talked to Livio and wrote about it here:

• Marcus Chown, The golden rule, The Guardian, 15 January 2003.

Chown writes:

Perhaps the most surprising place the golden ratio crops up is in the physics of black holes, a discovery made by Paul Davies of the University of Adelaide in 1989. Black holes and other self-gravitating bodies such as the sun have a “negative specific heat”. This means they get hotter as they lose heat. Basically, loss of heat robs the gas of a body such as the sun of internal pressure, enabling gravity to squeeze it into a smaller volume. The gas then heats up, for the same reason that the air in a bicycle pump gets hot when it is squeezed.

Things are not so simple, however, for a spinning black hole, since there is an outward “centrifugal force” acting to prevent any shrinkage of the hole. The force depends on how fast the hole is spinning. It turns out that at a critical value of the spin, a black hole flips from negative to positive specific heat—that is, from growing hotter as it loses heat to growing colder. What determines the critical value? The mass of the black hole and the golden ratio!

Why is the golden ratio associated with black holes? “It’s a complete enigma,” Livio confesses.

Extremal black holes

As we’ve seen, a rotating uncharged black hole has negative specific heat whenever the angular momentum is below a certain critical value:

\displaystyle{ J < k M^2 }


\displaystyle{ k = \sqrt{2 \sqrt{3} - 3} = 0.68125003863\dots }

As J goes up to this critical value, the specific heat actually approaches -\infty! On the other hand, a rotating uncharged black hole has positive specific heat when

\displaystyle{  J > kM^2}

and as J goes down to this critical value, the specific heat approaches -\infty. So, there’s some sort of ‘phase transition’ at

\displaystyle{  J = k M^2 }

But as we make the black hole spin even faster, something very strange happens when

\displaystyle{ J > M^2 }

Then the black hole gets a naked singularity!

In other words, its singularity is no longer hidden behind an event horizon. An event horizon is an imaginary surface such that if you cross it, you’re doomed to never come back out. As far as we know, all black holes in nature have their singularities hidden behind an event horizon. But if the angular momentum were too big, this would not be true!

A black hole posed right at the brink:

\displaystyle{ J = M^2 }

is called an ‘extremal’ black hole.

Black holes in nature

Most physicists believe it’s impossible for black holes to go beyond extremality. There are lots of reasons for this. But do any black holes seen in nature get close to extremality? For example, do any spin so fast that they have positive specific heat? It seems the answer is yes!

Over on Google+, Robert Penna writes:

Nature seems to have no trouble making black holes on both sides of the phase transition. The spins of about a dozen solar mass black holes have reliable measurements. GRS1915+105 is close to J=M^2. The spin of A0620-00 is close to J=0. GRO J1655-40 has a spin sitting right at the phase transition.

The spins of astrophysical black holes are set by a competition between accretion (which tends to spin things up to J=M^2) and jet formation (which tends to drain angular momentum). I don’t know of any astrophysical process that is sensitive to the black hole phase transition.

That’s really cool, but the last part is a bit sad! The problem, I suspect, is that Hawking radiation is so pathetically weak.

But by the way, you may have heard of this recent paper—about a supermassive black hole that’s spinning super-fast:

• G. Risaliti, F. A. Harrison, K. K. Madsen, D. J. Walton, S. E. Boggs, F. E. Christensen, W. W. Craig, B. W. Grefenstette, C. J. Hailey, E. Nardini, Daniel Stern and W. W. Zhang, A rapidly spinning supermassive black hole at the centre of NGC 1365, Nature (2013), 449–451.

They estimate that this black hole has a mass about 2 million times that of our sun, and that

\displaystyle{ J \ge 0.84 \, M^2 }

with 90% confidence. If so, this is above the phase transition where it gets positive specific heat.

The nitty-gritty details

Here is where Paul Davies claimed the golden ratio shows up in black hole physics:

• Paul C. W. Davies, Thermodynamic phase transitions of Kerr-Newman black holes in de Sitter space, Classical and Quantum Gravity 6 (1989), 1909–1914.

He works out when the specific heat vanishes for rotating and/or charged black holes in a universe with a positive cosmological constant: so-called de Sitter space. The formula is pretty complicated. Then he set the cosmological constant \Lambda to zero. In this case de Sitter space flattens out to Minkowski space, and his black holes reduce to Kerr–Newman black holes: that is, rotating and/or charged black holes in an asymptotically Minkowskian spacetime. He writes:

In the limit \alpha \to 0 (that is, \Lambda \to 0), the cosmological horizon no longer exists: the solution corresponds to the case of a black hole in asymptotically flat spacetime. In this case r may be explicitly eliminated to give

(\beta + \gamma)^3 + \beta^2 -\beta - \frac{3}{4} \gamma^2  = 0.   \qquad (2.17)


\beta = a^2 / M^2

\gamma = Q^2 / M^2

and he says M is the black hole’s mass, Q is its charge and a is its angular momentum. He continues:

For \beta = 0 (i.e. a = 0) equation (2.17) has the solution \gamma = 3/4, or

\displaystyle{ Q^2 = \frac{3}{4} M^2 } \qquad  (2.18)

For \gamma = 0 (i.e. Q = 0), equation (2.17) may be solved to give \beta = (\sqrt{5} - 1)/2 or

\displaystyle{ a^2 = (\sqrt{5} - 1)M^2/2 \cong 0.62 M^2   }  \qquad  (2.19)

These were the results first reported for the black-hole case in Davies (1979).

In fact a can’t be the angular momentum, since the right condition for a phase transition should say the black hole’s angular momentum is some constant times its mass squared. I think Davies really meant to define

a = J/M

This is important beyond the level of a mere typo, because we get different concepts of specific heat depending on whether we hold J or a constant while taking certain derivatives!

In the usual definition of specific heat for rotating black holes, we hold J constant and see how the black hole’s heat energy changes with temperature. If we call this specific heat C_J, we have

\displaystyle{ C_J = T \left.\frac{\partial S}{\partial T}\right|_J }

where S is the black hole’s entropy. This specific heat C_J becomes infinite when

\displaystyle{ \frac{J^2}{M^4} = 2 \sqrt{3} - 3  }

But if instead we hold a = J/M constant, we get something else—and this what Davies did! If we call this modified concept of specific heat C_a, we have

\displaystyle{ C_a = T \left.\frac{\partial S}{\partial T}\right|_a }

This modified ‘specific heat’ C_a becomes infinite when

\displaystyle{  \frac{J^2}{M^4} = \frac{\sqrt{5}-1}{2} }

After proving these facts in the comments below, Greg Egan drew some nice graphs to explain what’s going on. Here are the curves of constant temperature as a function of the black hole’s mass M and angular momentum J:

The dashed parabola passing through the peaks of the curves of constant temperature is where C_J becomes infinite. This is where energy can be added without changing the temperature, so long as it’s added in a manner that leaves J constant.

And here are the same curves of constant temperature, along with the parabola where C_a becomes infinite:

This new dashed parabola intersects each curve of constant temperature at the point where the tangent to this curve passes through the origin: that is, where the tangent is a line of constant a=J/M. This is where energy and angular momentum can be added to the hole in a manner that leaves a constant without changing the temperature.

As mentioned, Davies correctly said when the ordinary specific heat C_J becomes infinite in another paper, eleven years earlier:

• Paul C. W. Davies, Thermodynamics of black holes, Rep. Prog. Phys. 41 (1978), 1313–1355.

You can see his answer on page 1336.

This 1978 paper, in turn, is a summary of previous work including an article from a year earlier:

• Paul C. W. Davies, The thermodynamic theory of black holes, Proc. Roy. Soc. Lond. A 353 (1977), 499–521.

And in the abstract of this earlier article, Davies wrote:

The thermodynamic theory underlying black-hole processes is developed in detail and applied to model systems. It is found that Kerr-Newman black holes undergo a phase transition at an angular-momentum mass ratio of 0.68M or an electric charge (Q) of 0.86M, where the heat capacity has an infinite discontinuity. Above the transition values the specific heat is positive, permitting isothermal equilibrium with a surrounding heat bath.

Here the number 0.68 is showing up because

\displaystyle{ \sqrt{ 2 \sqrt{3} - 3 } = 0.68125003863\dots }

The number 0.86 is showing up because

\displaystyle{ \sqrt{ \frac{3}{4} } = 0.86602540378\dots }

By the way, just in case you want to do some computations using experimental data, let me put the speed of light c and gravitational constant G back in the formulas. A rotating (uncharged) black hole is extremal when

\displaystyle{ c J = G M^2 }

Milankovich vs the Ice Ages

30 January, 2013

guest post by Blake Pollard

Hi! My name is Blake S. Pollard. I am a physics graduate student working under Professor Baez at the University of California, Riverside. I studied Applied Physics as an undergraduate at Columbia University. As an undergraduate my research was more on the environmental side; working as a researcher at the Water Center, a part of the Earth Institute at Columbia University, I developed methods using time-series satellite data to keep track of irrigated agriculture over northwestern India for the past decade.

I am passionate about physics, but have the desire to apply my skills in more terrestrial settings. That is why I decided to come to UC Riverside and work with Professor Baez on some potentially more practical cross-disciplinary problems. Before starting work on my PhD I spent a year surfing in Hawaii, where I also worked in experimental particle physics at the University of Hawaii at Manoa. My current interests (besides passing my classes) lie in exploring potential applications of the analogy between information and entropy, as well as in understanding parallels between statistical, stochastic, and quantum mechanics.

Glacial cycles are one essential feature of Earth’s climate dynamics over timescales on the order of 100′s of kiloyears (kyr). It is often accepted as common knowledge that these glacial cycles are in some way forced by variations in the Earth’s orbit. In particular many have argued that the approximate 100 kyr period of glacial cycles corresponds to variations in the Earth’s eccentricity. As we saw in Professor Baez’s earlier posts, while the variation of eccentricity does affect the total insolation arriving to Earth, this variation is small. Thus many have proposed the existence of a nonlinear mechanism by which such small variations become amplified enough to drive the glacial cycles. Others have proposed that eccentricity is not primarily responsible for the 100 kyr period of the glacial cycles.

Here is a brief summary of some time series analysis I performed in order to better understand the relationship between the Earth’s Ice Ages and the Milankovich cycles.

I used publicly available data on the Earth’s orbital parameters computed by André Berger (see below for all references). This data includes an estimate of the insolation derived from these parameters, which is plotted below against the Earth’s temperature, as estimated using deuterium concentrations in an ice core from a site in the Antarctic called EPICA Dome C:

As you can see, it’s a complicated mess, even when you click to enlarge it! However, I’m going to focus on the orbital parameters themselves, which behave more simply. Below you can see graphs of three important parameters:

• obliquity (tilt of the Earth’s axis),
• precession (direction the tilted axis is pointing),
• eccentricity (how much the Earth’s orbit deviates from being circular).

You can click on any of the graphs here to enlarge them:

Richard Muller and Gordon MacDonald have argued that another astronomical parameter is important: the angle between the plane Earth’s orbit and the ‘invariant plane’ of the solar system. This invariant plane of the solar system depends on the angular momenta of the planets, but roughly coincides with the plane of Jupiter’s orbit, from what I understand. Here is a plot of the orbital plane inclination for the past 800 kyr:

One can see from these plots, or from some spectral analysis, that the main periodicities of the orbital parameters are:

• Obliquity ~ 42 kyr
• Precession ~ 21 kyr
• Eccentricity ~100 kyr
• Orbital plane ~ 100 kyr

Of course the curves clearly are not simple sine waves with those frequencies. Fourier transforms give information regarding the relative power of different frequencies occurring in a time series, but there is no information left regarding the time dependence of these frequencies as the time dependence is integrated out in the Fourier transform.

The Gabor transform is a generalization of the Fourier transform, sometimes referred to as the ‘windowed’ Fourier transform. For the Fourier transform:

\displaystyle{ F(w) = \dfrac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-iwt} \, dt}

one may think of e^{-iwt} , the ‘kernel function’, as the guy acting as your basis element in both spaces. For the Gabor transform instead of e^{-iwt} one defines a family of functions,

g_{(b,\omega)}(t) = e^{i\omega(t-b)}g(t-b)

where g \in L^{2}(\mathbb{R}) is called the window function. Typical windows are square windows and triangular (Bartlett) windows, but the most common is the Gaussian:

\displaystyle{ g(t)= e^{-kt^2} }

which is used in the analysis below. The Gabor transform of a function f(t) is then given by

\displaystyle{ G_{f}(b,w) = \int_{-\infty}^\infty f(t) \overline{g(t-b)} e^{-iw(t-b)} \, dt }

Note the output of a Gabor transform, like the Fourier transform, is a complex function. The modulus of this function indicates the strength of a particular frequency in the signal, while the phase carries information about the… well, phase.

For example the modulus of the Gabor transform of

\displaystyle{ f(t)=\sin(\dfrac{2\pi t}{100}) }

is shown below. For these I used the package Rwave, originally written in S by Rene Carmona and Bruno Torresani; R port by Brandon Whitcher.

You can see that the line centered at a frequency of .01 corresponds to the function’s period of 100 time units.

A Fourier transform would do okay for such a function, but consider now a sine wave whose frequency increases linearly. As you can see below, the Gabor transform of such a function shows the linear increase of frequency with time:

The window parameter in both of the above Gabor transforms is 100 time units. Adjusting this parameter effects the vertical blurriness of the Gabor transform. For example here is the same plot as a above, but with window parameters of 300, 200, 100, and 50 time units:

You can see as you make the window smaller the line gets sharper, but only to a point. When the window becomes approximately smaller than a given period of the signal the line starts to blur again. This makes sense, because you can’t know the frequency of a signal precisely at a precise moment in time… just like you can’t precisely know both the momentum and position of a particle in quantum mechanics! The math is related, in fact.

Now let’s look at the Earth’s temperature over the past 800 kyr, estimated from the EPICA ice core deuterium concentrations:

When you look at this, first you notice spikes occurring about every 100 kyr. You can also see that the last 5 of these spikes appear to be bigger and more dramatic than the ones occurring before 500 kyr ago. Roughly speaking, each of these spikes corresponds to rapid warming of the Earth, after which occurs slightly less rapid cooling, and then a slow decrease in temperature until the next spike occurs. These are the Earth’s glacial cycles.

At the bottom of the curve, where the temperature is about about 4 °C cooler than the mean of this curve, glaciers are forming and extending down across the northern hemisphere. The relatively warm periods on the top of the spikes, about 10 °C hotter than the glacial periods. are called the interglacials. You can see that we are currently in the middle of an interglacial, so the Earth is relatively warm compared to rest of the glacial cycles.

Now we’ll take a look at the windowed Fourier transform, or the Gabor transform, of this data. The window size for these plots is 300 kyr.

Zooming in a bit, one can see a few interesting features in this plot:

We see one line at a frequency of about .024, with a sampling rate of 1 kyr, corresponds to a period of about 42 kyr, close to the period of obliquity. We also see a few things going on around a frequency of .01, corresponding to a 100 kyr period.

The band at .024 appears to be relatively horizontal, indicating an approximately constant frequency. Around the 100 kyr periods there is more going on. At a slightly higher frequency, about .015, there appears to be a band of slowly increasing frequency. Also, around .01 it’s hard to say what is really going on. It is possible that we see a combination of two frequency elements, one increasing, one decreasing, but almost symmetric. This may just be an artifact of the Gabor transform or the window and frequency parameters.

The window size for the plots below is slightly smaller, about 250 kyr. If we put the temperature and obliquity Gabor Transforms side by side, we see this:

It’s clear the lines at .024 line up pretty well.

Doing the same with eccentricity:

Eccentricity does not line up well with temperature in this exercise though both have bright bands above and below .01 .

Now for temperature and orbital inclination:

One sees that the frequencies line up better for this than for eccentricity, but one has to keep in mind that there is a nonlinear transformation performed on the ‘raw’ orbital plane data to project this down into the ‘invariant plane’ of the solar system. While this is physically motivated, it surely nudges the spectrum.

The temperature data clearly has a component with a period of approximately 42 kyr, matching well with obliquity. If you tilt your head a bit you can also see an indication of a fainter response at a frequency a bit above .04, corresponding roughly to period just below 25 kyrs, close to that of precession.

As far as the 100 kyr period goes, which is the periodicity of the glacial cycles, this analysis confirms much of what is known, namely that we can’t say for sure. Eccentricity seems to line up well with a periodicity of approximately 100 kyr, but on closer inspection there seems to be some discrepancies if you try to understand the glacial cycles as being forced by variations in eccentricity. The orbital plane inclination has a more similar Gabor transform modulus than does eccentricity.

A good next step would be to look the relative phases of the orbital parameters versus the temperature, but that’s all for now.

If you have any questions or comments or suggestions, please let me know!


The orbital data used above is due to André Berger et al and can be obtained here:

Orbital variations and insolation database, NOAA/NCDC/WDC Paleoclimatology.

The temperature proxy is due to J. Jouzel et al, and it’s based on changes in deuterium concentrations from the EPICA Antarctic ice core dating back over 800 kyr. This data can be found here:

EPICA Dome C – 800 kyr deuterium data and temperature estimates, NOAA Paleoclimatology.

Here are the papers by Muller and Macdonald that I mentioned:

• Richard Muller and Gordan MacDonald, Glacial cycles and astronomical forcing, Science 277 (1997), 215–218.

• Richard Muller and Gordan MacDonald, Spectrum of 100-kyr glacial cycle: orbital inclination, not eccentricity, PNAS 1997, 8329–8334.

They also have a book:

• Richard Muller and Gordan MacDonald, Ice Ages and Astronomical Causes, Springer, Berlin, 2002.

You can also get files of the data I used here:

Berger et al orbital parameter data, with explanatory text here.

Jouzel et al EPICA Dome C temperature data, with explanatory text here.

Muller and Macdonald’s orbital plane inclination data.

Our Galactic Environment

27 December, 2012

While I’m focused on the Earth these days, I can’t help looking up and thinking about outer space now and then.

So, let me tell you about the Kuiper Belt, the heliosphere, the Local Bubble—and what may happen when our Solar System hits the next big cloud! Could it affect the climate on Earth?

New Horizons

We’re going on a big adventure!

New Horizons has already taken great photos of volcanoes on Jupiter’s moon Io. It’s already closer to Pluto than we’ve ever been. And on 14 July 2016 it will fly by Pluto and its moons Charon, Hydra, and Nix!

But that’s just the start: then it will go to see some KBOs!

The Kuiper Belt stretches from the orbit of Neptune to almost twice as far from the Sun. It’s a bit like the asteroid belt, but much bigger: 20 times as wide and 20 – 200 times as massive. But while most asteroids are made of rock and metal, most Kuiper Belt Objects or ‘KBOs’ are composed largely of frozen methane, ammonia and water.

The Earth’s orbit has a radius of one astronomical unit, or AU. The Kuiper Belt goes from 30 AU to 50 AU out. For comparison, the heliosphere, the region dominated by the energetic fast-flowing solar wind, fizzles out around 120 AU. That’s where Voyager 1 is now.

New Horizons will fly through the Kuiper Belt from 2016 to 2020… and, according to plan, its mission will end in 2026. How far out will it be then? I don’t know! Of course it will keep going…

For more see:

• JPL, New Horizons: NASA’s Pluto-Kuiper Belt Mission.

The heliosphere

Here’s a young star zipping through the Orion Nebula. It’s called LL Orionis, and this picture was taken by the Hubble Telescope in February 1995:

The star is moving through the interstellar gas at supersonic speeds. So, when this gas hits the fast wind of particles shooting out from the star, it creates a bow shock half a light-year across. It’s a bit like when a boat moves through the water faster than the speed of water waves.

There’s also a bow shock where the solar wind hits the Earth’s magnetic field. It’s about 17 kilometers thick, and located about 90,000 kilometers from Earth:

For a long time scientists thought there was a bow shock where nearby interstellar gas hit the Sun’s solar wind. But this was called into question this year when a satellite called the Interstellar Boundary Explorer (IBEX) discovered the Solar System is moving slower relative to this gas than we thought!

IBEX isn’t actually going to the edge of the heliosphere—it’s in Earth orbit, looking out. But Voyager 1 seems close to hitting the heliopause, where the Earth’s solar wind comes to a stop. And it’s seeing strange things!

The Interstellar Boundary Explorer

The Sun shoots out a hot wind of ions moving at 300 to 800 kilometers per second. They form a kind of bubble in space: the heliosphere. These charged particles slow down and stop when they hit the hydrogen and helium atoms in interstellar space. But those atoms can penetrate the heliosphere, at least when they’re neutral—and a near-earth satellite called IBEX, the Interstellar Boundary Explorer, has been watching them! And here’s what IBEX has seen:

In December 2008, IBEX first started detecting energetic neutral atoms penetrating the heliosphere. By October 2009 it had collected enough data to see the ‘IBEX ribbon’: an unexpected arc-shaped region in the sky has many more energetic neutral atoms than expected. You can see it here!

The color shows how many hundreds of energetic neutral atoms are hitting the heliosphere per second per square centimeter per keV. A keV, or kilo-electron-volt, is a unit of energy. Different atoms are moving with different energies, so it makes sense to count them this way.

You can see how the Voyager spacecraft are close to leaving the heliosphere. You can also see how the interstellar magnetic field lines avoid this bubble. Ever since the IBEX ribbon was detected, the IBEX team has been trying to figure out what causes it. They think it’s related to the interstellar magnetic field. The ribbon has been moving and changing intensity quite a bit in the couple of years they’ve been watching it!

Recently, IBEX announced that our Solar System has no bow shock—a big surprise. Previously, scientists thought the heliosphere created a bow-shaped shock wave in the interstellar gas as it moved along, like that star in the Orion Nebula we just looked at.

The Local Bubble

Get to know the neighborhood!

I love the names of these nearby stars! Some I knew: Vega, Altair, Fomalhaut, Alpha Centauri, Sirius, Procyon, Denebola, Pollux, Castor, Mizar, Aldebaran, Algol. But many I didn’t: Rasalhague, Skat, Gaorux, Pherkad, Thuban, Phact, Alphard, Wazn, and Algieba! How come none of the science fiction I’ve read uses these great names? Or maybe I just forgot.

The Local Bubble is a bubble of hot interstellar gas 300 light years across, probably blasted out by the supernova called Geminga near the bottom of this picture.


Here’s the sky viewed in gamma rays. A lot come from a blazar 7 billion light years away that erupted in 2005: a supermassive black hole at the center of a galaxy, firing particles in a jet that happens to be aimed straight at us. Some come from nearby pulsars: rapidly rotating neutron stars formed by the collapse of stars that went supernova. The one I want you to think about is Geminga.

Geminga is just 800 light years away from us, and it exploded only 300,000 years ago! That may seem far away and long ago to you, but not to me. The first Neanderthalers go back around 350,000 years… and they would have seen this supernova in the daytime, it was so close.

But here’s the reason I want you to think about Geminga. It seems to have blasted out the bubble of hot low-density gas our Solar System finds itself in: the Local Bubble. Astronomers have even detected micrometer-sized interstellar meteor particles coming from its direction!

We may think of interstellar space as all the same—empty and boring—but that’s far from true. The density of interstellar space varies immensely from place to place! The Local Bubble has just 0.05 atoms per cubic centimeter, but the average in our galaxy is about 20 times that, and we’re heading toward some giant clouds that are 2000 to 20,000 times as dense. The fun will start when we hit those…. but more on that later.

Nearby clouds

While we live in the Local Bubble, several thousand years ago we entered a small cloud of cooler, denser gas: the Local Fluff. We’ll leave this in at most 4 thousand years. But that’s just the beginning! As we pass the Scorpius-Centaurus Association, we’ll hit bigger, colder and denser clouds—and they’ll squash the heliosphere.

When will this happen? People seem very unsure. I’ve seen different sources saying we entered the Local Fluff sometime between 44,000 and 150,000 years ago, and that we’ll stay within it for between 4,000 and 20,000 years.

We’ll then return to the hotter, less dense gas of the Local Bubble until we hit the next cloud. That may take at least 50,000 years. Two candidates for the first cloud we’ll hit are the G Cloud and the Apex Cloud. The Apex Cloud is just 15 light years away:

• Priscilla C. Frisch, Local interstellar matter: the Apex Cloud.

When we hit a big cloud, it will squash the heliosphere. Right now, remember, this is roughly 120 AU in radius. But before we entered the Local Fluff, it was much bigger. And when we hit thicker clouds, it may shrink down to just 1 or 2 AU!

The heliosphere protects us from galactic cosmic rays. So, when we hit the next cloud, more of these cosmic rays will reach the Earth. Nobody knows for sure what the effects will be… but life on Earth has survived previous incidents like this, and other problems will hit us much sooner, so don’t stay awake at night worrying about it!

Indeed, ice core samples from the Antarctic show spikes in the concentration of the radioactive isotope beryllium-10 in two seperate events, one about 60,000 years ago and another about 33,000 years ago. These might have been caused by a sudden increase in cosmic rays. But nobody is really sure.

People have studied the possibility that cosmic rays could influence the Earth’s weather, for example by seeding clouds:

• K. Scherer, H. Fichtner et al, Interstellar-terrestrial relations: variable cosmic environments, the dynamic heliosphere, and their imprints on terrestrial archives and climate, Space Science Reviews 127 (2006), 327–465.

• Benjamin A. Laken, Enric Pallé, Jaša Čalogović and Eimear M. Dunne, A cosmic ray-climate link and cloud observations, J. Space Weather Space Clim. 2 (2012), A18.

Despite the title of the second paper, its conclusion is that “it is clear that there is no robust evidence of a widespread link between the cosmic ray flux and clouds.” That’s clouds on Earth, not clouds of interstellar gas! The first paper is much more optimistic about the existence of such a link, but it doesn’t provide a ‘smoking gun’.

And—in case you’re wondering—variations in cosmic rays this century don’t line up with global warming:

The top curves are the Earth’s temperature as estimated by GISTEMP (the brown curve), and the carbon dioxide concentration in the Earth’s atmosphere as measured by Charles David Keeling (in green). The bottom ones are galactic cosmic rays as measured by CLIMAX (the gray dots), the sunspot cycle as measured by the Solar Influences Data Analysis Center (in red), and total solar irradiance as estimated by Judith Lean (in blue).

But be careful: the galactic cosmic ray curve has been flipped upside down, since when solar activity is high, then fewer galactic cosmic rays make it to Earth! You can see that here:

I’m sorry these graphs aren’t neatly lined up, but you can see that peaks in the sunspot cycle happened near 1980, 1989 and 2002, which is when we had minima in the galactic cosmic rays.

For more on the neighborhood of the Solar System and what to expect as we pass through various interstellar clouds, try this great article:

• Priscilla Frisch, The galactic environment of the Sun, American Scientist 88 (January-February 2000).

I have lots of scientific heroes: whenever I study something, I find impressive people have already been there. This week my hero is Priscilla Frisch. She edited a book called Solar Journey: The Significance of Our Galactic Environment for the Heliosphere and Earth. The book isn’t free, but this chapter is:

• Priscilla C. Frisch and Jonathan D. Slavin, Short-term variations in the galactic environment of the Sun.

For more on how what the heliosphere might do when we hit the next big cloud, see:

• Hans-R. Mueller, Priscilla C. Frisch, Vladimir Florinski and Gary P. Zank, Heliospheric response to different possible interstellar environments.

The Aquila Rift

Just for fun, let’s conclude by leaving our immediate neighborhood and going a bit further out. Here’s a picture of the Aquila Rift, taken by Adam Block of the Mt. Lemmon SkyCenter at the University of Arizona:

The Aquila Rift is a region of molecular clouds about 600 light years away in the direction of the star Altair. Hundreds of stars are being formed in these clouds.

A molecular cloud is a region in space where the interstellar gas gets so dense that hydrogen forms molecules, instead of lone atoms. While the Local Fluff near us has about 0.3 atoms per cubic centimeter, and the Local Bubble is much less dense, a molecular cloud can easily have 100 or 1000 atoms per cubic centimeter. Molecular clouds often contain filaments, sheets, and clumps of submicrometer-sized dust particles, coated with frozen carbon monoxide and nitrogen. That’s the dark stuff here!

I don’t know what will happen to the Earth when our Solar System hits a really dense molecular cloud. It might have already happened once. But it probably won’t happen again for a long time.

Mathematics of the Environment (Part 9)

27 November, 2012

I didn’t manage to cover everything I intended last time, so I’m moving the stuff about the eccentricity of the Earth’s orbit to this week, and expanding it.

Sunshine and the Earth’s orbit

I bet some of you are hungry for some math. As I mentioned, it takes some work to see how changes in the eccentricity of the Earth’s orbit affect the annual average of sunlight hitting the top of the Earth’s atmosphere. Luckily Greg Egan has done this work for us. While the result is surely not new, his approach makes nice use of the fact that both gravity and solar radiation obey an inverse-square law. That’s pretty cool.

Here is his calculation with some details filled in.

Let’s think of the Earth as moving around an ellipse with one focus at the origin. Its angular momentum is then

\displaystyle{ J = m r v_\theta }

where m is its mass, r and \theta are its polar coordinates, and v_\theta is the angular component of its velocity:

\displaystyle{ v_\theta = r \frac{d \theta}{d t} }


\displaystyle{ J = m r^2 \frac{d \theta}{d t}  }


\displaystyle{\frac{d \theta}{d t} = \frac{J}{m r^2} }

Since the brightness of a distant object goes like 1/r^2, the solar energy hitting the Earth per unit time is

\displaystyle{ \frac{d U}{d t} = \frac{C}{r^2}}

for some constant C. It follows that the energy delivered per unit of angular progress around the orbit is

\displaystyle{ \frac{d U}{d \theta} = \frac{d U/d t}{d \theta/ dt} = \frac{C m}{J} }

Thus, the total energy delivered in one period will be

\begin{array}{ccl} U &=& \displaystyle{ \int_0^{2 \pi} \frac{d U}{d \theta}  \, d \theta} \\ \\ &=& \displaystyle{ \frac{2\pi C m}{J} }  \end{array}

So far we haven’t used the the fact that the Earth’s orbit is elliptical. Next we’ll do that. Our goal will be to show that U depends only very slightly on the eccentricity of the Earth’s orbit. But we need to review a bit of geometry first.

The geometry of ellipses

If the Earth is moving in an ellipse with one focus at the origin, its equation in polar coordinates is

\displaystyle{ r = \frac{p}{1 + e \cos \theta} }

where e is the eccentricity and p is the somewhat dirty-sounding semi-latus rectum. You can think of p as a kind of average radius of the ellipse—more on that in a minute.

Let’s think of the origin in this coordinate system as the Sun—that’s close to true, though the Sun moves a little. Then the Earth gets closest to the Sun when \cos \theta is as big as possible. So, the Earth is closest to the Sun when \theta = 0, and then its distance is

\displaystyle{ r_1 = \frac{p}{1 + e} }

Similarly, the Earth is farthest from the Sun happens when \theta = \pi, and then its distance is

\displaystyle{ r_2 = \frac{p}{1 - e} }

We call r_1 the perihelion and r_2 the aphelion.

The semi-major axis is half the distance between the opposite points on the Earth’s orbit that are farthest from each other. This is denoted a. These points occur at \theta = 0 and \theta = \pi, so the distance between these points is r_1 + r_2, and

\displaystyle{  a = \frac{r_1 + r_2}{2} }

So, the semi-major axis is the arithmetic mean of the perihelion and aphelion.

The semi-minor axis is half the distance between the opposite points on the Earth’s orbit that are closest to each other. This is denoted b.

Puzzle 1. Show that the semi-minor axis is the geometric mean of the perihelion and aphelion:

\displaystyle{  b = \sqrt{r_1 r_2} }

I said the semi-latus rectum p is also a kind of average radius of the ellipse. Just to make that precise, try this:

Puzzle 2. Show that the semi-latus rectum is the harmonic mean of the perihelion and aphelion:

\displaystyle{ p = \frac{1}{\frac{1}{2}\left(\frac{1}{r_1} + \frac{1}{r_2}\right) } }

This puzzle is just for fun: the Greeks loved arithmetic, geometric and harmonic means, and the Greek mathematician Apollonius wrote a book on conic sections, so he must have known these facts and loved them. The conventional wisdom is that the Greeks never realized that the planets move in elliptical orbits. However, the wonderful movie Agora presents a great alternative history in which Hypatia figures it all out shortly before being killed! And the mathematician Sandro Graffi (who incidentally taught a course I took in college on the self-adjointness of quantum-mechanical Hamiltonians) has claimed:

Now an infrequently read work of Plutarch, several parts of the Natural History of Plinius, of the Natural Questions of Seneca, and of the Architecture of Vitruvius, also infrequently read, especially by scientists, clearly show that the cultural elite of the early imperial age (first century A.D.) were fully aware of and convinced of a heliocentric dynamical theory of planetary motions based on the attractions of the planets toward the Sun by a force proportional to the inverse square of the distance between planet and Sun. The inverse square dependence on the distance comes from the assumption that the attraction is propagated along rays emanating from the surfaces of the bodies.

I have no idea if the controversial last part of this claim is true. But it’s fun to imagine!

More importantly for what’s to come, we can express the semi-minor axis in terms of the semi-major axis and the eccentricity. Since

\displaystyle{ r_1 = \frac{p}{1 + e} , \qquad r_2 = \frac{p}{1 - e} }

we have

\displaystyle{ r_1 + r_2 = \frac{p}{1 + e} + \frac{p}{1 - e} = \frac{2 p}{1 - e^2} }

so the semi-minor axis is

\displaystyle{ a = \frac{p}{1 - e^2} }


\displaystyle {r_1 r_2 = \frac{p^2}{1 - e^2} }

so the semi-major axis is

\displaystyle { b = \frac{p}{\sqrt{1 - e^2}} }

and thus they are related by

b = a \sqrt{1 - e^2}

Remember this!

How total annual sunshine depends on eccentricity

We saw a nice formula for the total solar energy hitting the Earth in one year in terms of its angular momentum J:

\displaystyle{ U = \frac{2\pi C m}{J} }

How can we relate the angular momentum J to the shape of the Earth’s orbit? The Earth’s energy, kinetic plus potential, is constant throughout the year. The kinetic energy is

\frac{1}{2}m v^2

and the potential energy is

\displaystyle{ -\frac{G M m}{r} }

At the aphelion or perihelion the Earth isn’t moving in or out, just around, so by our earlier work

\displaystyle{v = v_\theta = \frac{J}{m r} }

and the kinetic energy is

\displaystyle{ \frac{J^2}{2 r^2} }

Equating the Earth’s energy at aphelion and perihelion, we thus get

\displaystyle{\frac{J^2}{2m r_1^2} -\frac{G M m}{r_1} = \frac{J^2}{2m r_2^2} -\frac{G M m}{r_2} }

and doing some algebra:

\displaystyle{\frac{J^2}{2m} \left(\frac{1}{r_1^2} - \frac{1}{r_2^2}\right) =  G M m \left( \frac{1}{r_1} - \frac{1}{r_2} \right) }

\displaystyle{\frac{J^2}{2m} \left(\frac{r_2^2 - r_1^2}{r_1^2 r_2^2}\right) =  G M m \left( \frac{r_2 - r_1}{r_1 r_2} \right) }

\displaystyle{\frac{J^2}{2m} \left(\frac{r_1 + r_2}{r_1 r_2}\right) =  G M m }

and solving for J,

\displaystyle{ J = m \sqrt{\frac{2 G M r_1 r_2}{r_1 + r_2}} }

But remember that the semi-major and semi-minor axis of the Earth’s orbit are given by

\displaystyle{ a=\frac{1}{2} (r_1+r_2)} , \qquad \displaystyle{ b=\sqrt{r_1 r_2} }

respectively! So, we have

\displaystyle{ J = mb \sqrt{\frac{GM}{a}} }

This lets us rewrite our old formula for the energy U in the form of sunshine that hits the Earth each year:

\displaystyle{ U=\frac{2\pi C m}{J} = \frac{2\pi C}{b} \sqrt{\frac{a}{G M}} }

But we’ve also seen that

b = a \sqrt{1 - e^2}

so we get the formula we’ve been seeking:

\displaystyle{U=\frac{2\pi C}{\sqrt{G M a (1-e^2)}}}

This tells us U as a function of semi-major axis and eccentricity.

As we’ll see later, the semi-major axis a is almost unchanged by small perturbations of the Earth’s orbit. The main thing that changes is the eccentricity e. But if e is small, e^2 is even smaller, so U doesn’t change much when we change e.

We can make this more quantiative. Let’s work out how much the actual changes in the Earth’s orbit affect the amount of solar radiation it gets! As we’ll see, the semi-major axis is almost constant, so we can ignore that. Complicated calculations we can’t redo here show that the eccentricity varies between 0.005 and 0.058. We’ve seen the total energy the Earth gets each year from solar radiation is proportional to

\displaystyle{ \frac{1}{\sqrt{1-e^2}} }

When the eccentricity is at its lowest value, e = 0.005, we get

\displaystyle{ \frac{1}{\sqrt{1-e^2}} = 1.0000125 }

When the eccentricity is at its highest value, e = 0.058, we get

\displaystyle{\frac{1}{\sqrt{1-e^2}} = 1.00168626 }

So, the solar power hitting the Earth each year changes by a factor of

\displaystyle{1.00168626/1.0000125 = 1.00167373 }

In other words, it changes by merely 0.167%.

That’s very small And the effect on the Earth’s temperature would naively be even less!

Naively, we can treat the Earth as a greybody: an ideal object whose tendency to absorb or emit radiation is the same at all wavelengths and temperatures. Since the temperature of a greybody is proportional to the fourth root of the power it receives, a 0.167% change in solar energy received per year corresponds to a percentage change in temperature roughly one fourth as big. That’s a 0.042% change in temperature. If we imagine starting with an Earth like ours, with an average temperature of roughly 290 kelvin, that’s a change of just 0.12 kelvin!

The upshot seems to be this: in a naive model without any amplifying effects, changes in the eccentricity of the Earth’s orbit would cause temperature changes of just 0.12 °C!

This is much less than the roughly 5 °C change we see between glacial and interglacial periods. So, if changes in eccentricity are important in glacial cycles, we have some explaining to do. Possible explanations include season-dependent phenomena and climate feedback effects, like the ice albedo effect we’ve been discussing. Probably both are very important!

Adiabatic invariance

Why does the semi-major axis of the Earth’s orbit remain almost unchanged under small perturbations? The reason is that it’s an ‘adiabatic invariant’. This is basically just a fancy way of saying it remains almost unchanged. But the point is, there’s a whole theory of adiabatic invariants… which supposedly explains the near-constancy of the semi-major axis.

According to Wikipedia:

The Earth’s eccentricity varies primarily due to interactions with the gravitational fields of Jupiter and Saturn. As the eccentricity of the orbit evolves, the semi-major axis of the orbital ellipse remains unchanged. From the perspective of the perturbation theory used in celestial mechanics to compute the evolution of the orbit, the semi-major axis is an adiabatic invariant. According to Kepler’s third law the period of the orbit is determined by the semi-major axis. It follows that the Earth’s orbital period, the length of a sidereal year, also remains unchanged as the orbit evolves. As the semi-minor axis is decreased with the eccentricity increase, the seasonal changes increase. But the mean solar irradiation for the planet changes only slightly for small eccentricity, due to Kepler’s second law.

Unfortunately, even though I understand a bit about the general theory of adiabatic invariants, I have not gotten around to convincing myself that the semi-major axis is such a thing, for the perturbations experienced by the Earth.

Here’s something easier: checking that the semi-major axis of the Earth’s orbit determines the period of the Earth’s orbit, say T. To do this, first relate the angular momentum to the period by integrating the rate at which orbital area is swept out by the planet:

\displaystyle{\frac{1}{2}  r^2 \frac{d \theta}{d t} = \frac{J}{2 m} }

over one orbit. Since the area of an ellipse is \pi a b, this gives us:

\displaystyle{ J = \frac{2 \pi a b m}{T} }

On the other hand, we’ve seen

\displaystyle{J = m b \sqrt{\frac{G M}{a}}}

Equating these two expressions for J shows that the period is:

\displaystyle{ T = 2 \pi \sqrt{\frac{a^3}{G M}}}

So, the period depends only on the semi-major axis, not the eccentricity. Conversely, we could solve this equation to see that the semi-major axis depends only on the period, not the eccentricity.

I’m treating G and M as constants here. If the mass of the Sun decreases, as it eventually will when it becomes a red giant and puffs out lots of gas, the semi-major axes of the Earth’s orbit will change. It will actually increase! This is one reason people are still arguing about just when the Earth will get swallowed up by the Sun:

• David Appell, The Sun will eventually engulf the Earth—maybe, Scientific American, 8 September 2008.

And, to show just how subtle these things are, if the mass of the Sun slowly changes, while the semi-major axis of the Earth’s orbit will change, the eccentricity will remain almost unchanged. Why? Because for this kind of process, it’s the eccentricity that’s an adiabatic invariant!

Indeed, I got all excited when I started reading a homework problem in Landau and Lifschitz’s book Classical Mechanics, which describes adiabatic invariants for the gravitational 2-body problem. But I was bummed out when they concluded that the eccentricity was an adiabatic invariant for gradual changes in M. They didn’t discuss any problems for which the semi-major axis was an adiabatic invariant.

I’ll have to get back to this later sometime, probably with the help of a good book on celestial mechanics. If you’re curious about the concept of adiabatic invariant, start here:

Adiabatic invariant, Wikipedia.

and then try this:

• Marko Robnik, Theory of adiabatic invariants, February 2004.

And if you know how to show the Earth’s semi-major axis is an adiabatic invariant, please tell me how!

Wind and Water on Mars

11 November, 2012

Frosty dunes


I love this photo, because it shows that Mars is a lively place with wind and water. These dunes near the north pole, occupying a region the size of Texas, have been sculpted by wind into long lines with crests 500 meters apart. Their hollows are covered with frost, which appears bluish-white in this infrared photograph. The big white spot near the bottom is a hill 100 meters high.

For more info, go here:

• THEMIS, North polar sand sea.

If you download the full-sized version of this photo, either by clicking on my picture or going to this webpage, you’ll see it’s astoundingly detailed!

THEMIS is the Thermal Emission Imaging System aboard the Mars Odyssey spacecraft, which has been orbiting Mars since 2002. It combines a 5-wavelength visual imaging system with a 9-wavelength infrared imaging system. It’s been taking great pictures—especially of regions that are too rugged for rovers like Opportunity, Spirit and Curiosity.

Because those rovers landed in places that were chosen to be safe, the pictures they take sometimes make Mars look… well, a bit dull. It’s not!

Let me show you what I mean.



These are barchans on Mars, C-shaped sand dunes that slowly move through the desert like this:


And see the dark fuzzy stuff? More on that later!

Barchans are also found on Earth, and surely on many other planets across the Universe. They’re one of several basic dune patterns—an inevitable consequence of the laws of nature under fairly common conditions.

Sand gradually accumulates on the upwind side of a barchan. Then it falls down the other side, called the ‘slip face’. The upwind slope is gentle, while the slope of the slip face is the angle of repose for sand: the maximum angle it can tolerate before it starts slipping down. On Earth that’s between 32 and 34 degrees.

Puzzle: What is the angle of repose of sand on Mars? Does the weaker pull of gravity let sandpiles be steeper? Or are they just as steep as on Earth?

Barchans gradually migrate in the direction of the wind, with small barchans moving faster than big ones. And when barchans collide, the smaller ones pass right through the big ones! So, they’re a bit like what physicists call solitons: waves that maintain their identity like particles. However, they display more complicated behaviors.

This simulation shows what can happen when two collide:

Depending on the parameters, they can:

c: coalesce into one barchan,

b: breed to form more barchans,

bu: bud, with the smaller one splitting in two, or

s: act like solitons, with one going right through the other!

This picture is from here:

• Orencio Durán, Veit Schwámmle and Hans J. Herrmann, Simulations of binary collisions of barchan dunes: the collision dynamics and its influence on the dune size distribution.

In this picture there is no ‘offset’ between the colliding barchans: they hit head-on. With an offset, more complicated things can happen – check out this picture:

It may seem surprising that there’s enough wind on Mars to create dunes. After all, the air pressure there is about 1% what it is here on Earth! But in fact the wind speed on Mars often exceeds 200 kilometers per hour, with gusts up to 600 kilometers per hour. There are dust storms on Mars so big they were first seen from telescopes on Earth long ago. So, wind is a big factor in Martian geology:

• NASA, Mars exploration program: dust storms.

The Mars rover Spirit even got its solar panels cleaned by some dust devils, and it took some movies of them:



This picture shows a dune field less than 400 kilometers from the north pole, bordered on both sides by flat regions—but also a big cliff at one end.

Here’s a closeup of those dunes… with stands of trees on top?!?

No, that’s an optical illusion. But whatever it is, it’s something strange. Robert Krulwich put it nicely:

They were first seen in 1998; they don’t look like anything we have here on Earth. To this day, no one is sure what they are, but we now know this: They come, then they go. Every Martian spring, they appear out of nowhere, showing up—70 percent of the time—where they were the year before. They pop up suddenly, sometimes overnight. When winter comes, they vanish.

In 2010, astronomer Candy Hansen tried to explain what’s going on, writing:

There is a vast region of sand dunes at high northern latitudes on Mars. In the winter, a layer of carbon dioxide ice covers the dunes, and in the spring as the sun warms the ice it evaporates. This is a very active process, and sand dislodged from the crests of the dunes cascades down, forming dark streaks.

She focused our attention on this piece of the image:

and she wrote:

In the subimage falling material has kicked up a small cloud of dust. The color of the ice surrounding adjacent streaks of material suggests that dust has settled on the ice at the bottom after similar events.

Also discernible in this subimage are polygonal cracks in the ice on the dunes (the cracks disappear when the ice is gone).

More recently, though, scientists have suggested that geysers are involved in this process, which might make it very active indeed!

Geysers formed as frozen carbon dioxide turns to gas, shooting out clumps of dark, basaltic sand, which slide down the dunes… that’s the most popular explanation. But maybe they’re colonies of photosynthetic Martian microorganisms soaking up the sunlight! Or maybe geysers are shooting up dark stuff that’s organic matter formed by some biological process. A bunch form right around sunrise, so something is being rapidly triggered by the sun.

This has some nice prose and awesome pictures:

• Robert Krulwich, Are those spidery black things on Mars dangerous? (maybe), Krulwich Wonders, National Public Radio, 3 October 2012.

The big picture above, and Candy Hansen’s explanation, can be found here:

• HiRiSE, Falling material kicks up cloud of dust on dunes.

HiRiSE, which stands for High Resolution Imaging Science Experiments, is a project based in Arizona that’s created an amazing website full of great Mars photos. For more clues, try this:

Martian geyser, Wikipedia.

Cryptic terrain

What’s going on in this region of Mars?

Candy Hansen writes:

There is an enigmatic region near the south pole of Mars known as the “cryptic” terrain. It stays cold in the spring, even as its albedo darkens and the sun rises in the sky.

This region is covered by a layer of translucent seasonal carbon dioxide ice that warms and evaporates from below. As carbon dioxide gas escapes from below the slab of seasonal ice it scours dust from the surface. The gas vents to the surface, where the dust is carried downwind by the prevailing wind.

The channels carved by the escaping gas are often radially organized and are known informally as “spiders.”

This is from:

• HiRISE, Cryptic terrain on Mars.

Vastitas Borealis

Here’s ice in a crater in the northern plains on Mars—the region with the wonderful name Vastitas Borealis:

Many scientists believe this huge plain was an ocean during the Hesperian Epoch, a period of Martian history that stretches from about 3.5 to about 1.8 billion years ago. Later, around the end of the Hesperian, they think about 30% of the water on Mars evaporated and left the atmosphere, drifting off into outer space… part of the danger of life on a planet without much gravity. The oceans then froze. Most of them slowly sublimated, disappearing into water vapor without ever melting. This water vapor was also lost to outer space.

• Linda M. V. Martel, Ancient floodwaters and seas on Mars.

But there’s still a lot of water left, especially in the polar ice caps. The north pole has an ice cap with 820,000 cubic kilometers of ice! That’s equal to 30% of the Earth’s Greenland ice sheet—enough to cover the whole surface of Mars to a depth of 5.6 meters if it melted, if we pretend Mars is flat.

And the south pole is covered by a slab of ice about 3 kilometers thick, a mixture of 85% carbon dioxide ice and 15% water ice, surrounded by steep slopes made almost entirely of water ice. This has enough water that if it melted it would cover the whole surface to a depth of 11 meters!

There’s also lots of permafrost underground, and frost on the surface, and bits of ice like this. The picture above was taken by the Mars Express satellite:

• ESA, Water ice in crater at Martian north pole.

The image is close to natural color, but the vertical relief is exaggerated by a factor of 3. The crater is 35 kilometers wide and 2 kilometers deep. It’s incredible how they can get this kind of picture from satellite photos and lots of clever image processing. I hope they didn’t do too much stuff just to make it look pretty.

Chasma Boreale

Here is the north pole of Mars:

As in Antarctica and Greenland, cold dense air flows downwards off the polar ice cap, creating intense winds called katabatic winds. These pick up and redeposit surface ice to make grooves in the ice. The swirly pattern comes from the Coriolis effect: while the winds are blowing more or less straight, Mars is turning around its pole, so they seem to swerve.

As you can see, the north polar ice cap has a huge canyon running through it, called Chasma Boreale:

Here’s an amazing picture of what it’d be like to stand near the head of this chasm:

Click to enlarge this—it deserves to be bigger! Here’s the story:

Climatic cycles of ice and dust built the Martian polar caps, season by season, year by year—and then whittled down their size when the climate changed. Here we are looking at the head of Chasma Boreale, a canyon that reaches 570 kilometers (350 miles) into the north polar cap. Canyon walls rise about 1,400 meters (4,600 feet) above the floor. Where the edge of the ice cap has retreated, sheets of sand are emerging that accumulated during earlier ice-free climatic cycles. Winds blowing off the ice have pushed loose sand into dunes, then driven them down-canyon in a westward direction, toward our viewpoint.

The above picture was cleverly created using photos from THEMIS. The vertical scale has been exaggerated by a factor of 2.5, I’m sad to say. You can download a 9-megabyte version from here:

• THEMIS, Chasma Boreale and the north polar ice cap.

and you can see an actual photo of this same canyon here:

• THEMIS, Dunes and ice in Chasma Boreale.

It’s beautifully detailed; here’s a miniature version:

and a sub-image that shows the layers of ice and sand:

Scientists are studying these layers in the ice cap to see if they match computer simulations of the climate of Mars. Just as the Earth’s orbit goes through changes called Milankovitch cycles, so does the orbit of Mars. These affect the climate: for example, when the tilt is big the tropics become colder, and polar ice migrates toward the equator. I don’t know much about this, despite my interest in Milankovitch cycles. What’s a good place to start learning more?

Here’s a closer view of icy dunes near the North pole:

Martian Sunset

As we’ve seen, Mars is a beautiful world, but a world in a minor key, a world whose glory days—the Hesperian Epoch—are long gone, whose once grand oceans are now reduced to windy canyons, icy dunes, and the massive ice caps of the poles. Let’s say goodbye to it for now… leaving off with this Martian sunset, photographed by the rover Spirit in Gusev Crater on May 19th, 2005.

• NASA Mars Exploration Rover Mission, A moment frozen in time.

This Panoramic Camera (Pancam) mosaic was taken around 6:07 in the evening of the rover’s 489th martian day, or sol. Spirit was commanded to stay awake briefly after sending that sol’s data to the Mars Odyssey orbiter just before sunset. This small panorama of the western sky was obtained using Pancam’s 750-nanometer, 530-nanometer and 430-nanometer color filters. This filter combination allows false color images to be generated that are similar to what a human would see, but with the colors slightly exaggerated. In this image, the bluish glow in the sky above the Sun would be visible to us if we were there, but an artifact of the Pancam’s infrared imaging capabilities is that with this filter combination the redness of the sky farther from the sunset is exaggerated compared to the daytime colors of the martian sky.

Because Mars is farther from the Sun than the Earth is, the Sun appears only about two-thirds the size that it appears in a sunset seen from the Earth. The terrain in the foreground is the rock outcrop “Jibsheet”, a feature that Spirit has been investigating for several weeks (rover tracks are dimly visible leading up to Jibsheet). The floor of Gusev crater is visible in the distance, and the Sun is setting behind the wall of Gusev some 80 km (50 miles) in the distance.

This mosaic is yet another example from MER of a beautiful, sublime martian scene that also captures some important scientific information. Specifically, sunset and twilight images are occasionally acquired by the science team to determine how high into the atmosphere the martian dust extends, and to look for dust or ice clouds. Other images have shown that the twilight glow remains visible, but increasingly fainter, for up to two hours before sunrise or after sunset. The long martian twilight (compared to Earth’s) is caused by sunlight scattered around to the night side of the planet by abundant high altitude dust. Similar long twilights or extra-colorful sunrises and sunsets sometimes occur on Earth when tiny dust grains that are erupted from powerful volcanoes scatter light high in the atmosphere.

Rolling Circles and Balls (Part 3)

11 September, 2012

In Part 1 and Part 2 we looked at the delightful curves you get by rolling one circle on another. Now let’s see what happens when you roll one circle inside another!

Four times as big

If you roll a circle inside a circle that’s 4 times as big, we get an astroid:

Puzzle 1. How many times does the rolling circle turn as it rolls all the way around?

By the way: don’t confuse an astroid with an asteroid. They both got their names because someone thought they looked like stars, but that’s where resemblance ends!

You can get an astroid using this funny parody of the equation for a circle:

x^{2/3} + y^{2/3} = 1

Or, if you don’t like equations, you can get a quarter of an astroid by letting a ladder slide down a wall and taking a time-lapse photo!

In other words, you get a whole astroid by taking the envelope of all line segments of length 1 going from some point on the x axis to some point on the y axis!

Three times as big

If the fixed circle is just 3 times as big as the one rolling inside it, we get an deltoid:

Puzzle 2. Now how many times does the rolling circle turn as it rolls all the way around?

By the way: it looks like we’re back to naming curves after body parts… but we’re not: both this curve and the muscle called a deltoid got their names because they look like the Greek letter delta:

Puzzle 3. Did the Greek letter delta get that name because it was shaped like a river delta, or was it the other way around?

As you might almost expect by now, if you’ve been reading this whole series, there are weird relations between the deltoid and the astroid.

For example: take a deltoid and shine parallel rays of light at it from any direction. Then the envelope of these rays is an astroid!

We summarize this by saying that the astroid is a catacaustic of the deltoid. This picture is by Xah Lee, who has also made a nice movie of what happens as you rotate the light source:

• Xah Lee, Deltoid catacaustic movie.

I don’t completelly understand the rays going through the deltoid, in either the picture or the movie. It looks like those rays are getting refracted, but that would be a diacaustic, not a catacaustic. ,I think they’re formed by continuing reflected rays to straight lines that go through the deltoid. If you didn’t do that you wouldn’t get a whole astroid, just part of one.

Anyway, at the very least you get part of an astroid, which you can complete to a whole one. And then, as you rotate the light source, the astroid you get rolls around the deltoid in a pleasant manner! This is nicely illustrated here:

Deltoid catacaustic, Wolfram Mathworld.

You can also get a deltoid from a deltoid! Draw all the osculating circles of the deltoid—that is, circles that match the deltoid’s curvature as well as its slope at the points they touch. The centers of these circles lie on another, larger deltoid:

We summarize this by saying that the evolute of a deltoid is another deltoid.

There are also fancier ways to get deltoids if you know more math. For example, the set of traces of matrices lying in the group SU(3) forms a filled-in deltoid in the complex plane!

This raises a question which unfortunately I’m too lazy to answer myself. So, I’ll just pose it as a puzzle:

Puzzle 4. Is the set of traces of matrices lying in SU(4) a filled-in astroid? In simpler terms, consider the values of a + b + c + d that we can get from complex numbers a,b,c,d with |a| = |b| = |c| = |d| = 1 and abcd = 1. Do these values form a filled-in astroid?

Twice as big

But now comes the climax of today’s story: what happens when we let a circle roll inside a circle that’s exactly twice as big?

Now you can see the rolling circle turn around once as it rolls all the way around the big one.

More excitingly, if we track a point on the rolling circle, it traces out a straight line! Can you see why? Later I’ll give a few proofs.

This gadget is called a Tusi couple. You could use it to convert a rolling motion into a vibrating one using some gears. Greg Egan made a nice animation showing how:

The Tusi couple is named after the Persian astronomer and mathematician Nasir al-Din al-Tusi, who discovered it around 1247, when he wrote a commentary on Ptomely’s Almagest, an important astronomical text:

He wrote:

If two coplanar circles, the diameter of one of which is equal to half the diameter of the other, are taken to be internally tangent at a point, and if a point is taken on the smaller circle—and let it be at the point of tangency—and if the two circles move with simple motions in opposite direction in such a way that the motion of the smaller is twice that of the larger so the smaller completes two rotations for each rotation of the larger, then that point will be seen to move on the diameter of the larger circle that initially passes through the point of tangency, oscillating between the endpoints.

I don’t quite understand why he was interested in this, but it has something to do with using epicycles to build linear motion out of circular motion. It also has something to do with the apparent motion of planets between the Earth and the Sun.

Later Copernicus also studied the Tusi couple. He proved that the moving point really did trace out a straight line:

However, many suspect that this was not a true rediscovery: al-Tusi had also proved this, and some aspects of Copernicus’ proof seem too similar to al-Tusi’s to be coincidence:

• George Saaliba, Whose science is Arabic science in Renaissance Europe?, Section 2: Arabic/Islamic science and the Renaissance science in Italy.

• I. N. Veselovsky, Copernicus and Nasir al-Din al-Tusi, Journal for the History of Astronomy 4 (1973), 128–130.

In fact, the Tusi couple goes back way before al-Tusi. It was known to Proclus back around 450 AD! Apparently he wrote about it in his Commentary on the First Book of Euclid. Proclus is mainly famous as a philosopher: people think of him as bringing neo-Platonism to its most refined heights. Given that, it’s no surprise that he also liked math. Indeed, he said:

Wherever there is number, there is beauty.

And of course this is what I’ve been trying to show you throughout this series.

Why the Tusi couple works

So, here are three proofs that a Tusi couple really does trace out a straight line. I posed this as a puzzle on Google+, and here are my favorite three answers. I like them because they’re very different. Since people have thought about Tusi couples since 450 AD, I doubt any of these proofs are original to the people I’m mentioning here! Still, they deserve credit.

The first, due to Omar Antolín Camarena, is in the style of traditional Euclidean geometry.

Let O be the center of the big circle. Let A be the position of the traced point at the instant t when it’s the point of tangency between the circles, let B be its position at some future time t′ and let C be the point of tangency at that same time t′. Let X be the center of the small circle at that future time t′.

We want to prove A, B and O lie on a line. To do this, it suffices to show that the angle AOC equals the angle BOC:

Want: ∠BOC = ∠AOC

The arc AC of the big circle has the same length as the arc BC of the small circle, since they are both the distance rolled between times t and t′. But the big circle is twice as, so the angle BXC on the little circle must be twice the angle AOC on the big circle:

Know: ∠BXC = 2 ∠AOC

But it’s a famous fact in Euclidean geometry that the angle BXC is twice the angle BOC:

Know: ∠BXC = 2 ∠BOC

From the two equations we know, the one we want follows!

Here’s the proof of that ‘famous fact’, in case you forgot it:

The second proof is due to Greg Egan. He distilled it down to a moving picture:

It’s a ‘proof without words’, so you may need to think a while to see how it shows that the Tusi couple traces out a straight line.

The third proof is due to Boris Borcic. It uses complex numbers. Let the big circle be the unit circle in the complex plane. Then the point of contact between the rolling circle and the big one is:

e^{i t}

so the center of the rolling circle is:

\displaystyle{ \frac{e^{it}}{2} }

Since the rolling circle turns around once clockwise as it rolls around the big one, the point whose motion we’re tracking here:

is equal to:

\displaystyle{ \frac{e^{it}}{2} + \frac{e^{i(\pi - t)}}{2} =  \frac{e^{i t} - e^{-i t}}{2}  = i \sin t =  i \; \mathrm{Im}(e^{it}) }

So, this point moves up and down along a vertical line, and its height equals the height of the point of contact, e^{it}.

But the same sort of argument shows that if we track the motion of the opposite point on the rolling circle, it

\displaystyle{ \frac{e^{it}}{2} - \frac{e^{i(\pi - t)}}{2} =  \frac{e^{i t} + e^{-i t}}{2}  =  \cos t =   \mathrm{Re}(e^{it}) }

So, this opposite point moves back and forth along a horizontal straight line… and its horizontal coordinate equals that of the point of contact!

You can see all this clearly in the animation Borcic made:

The point of contact, the tip of red arrowhead, is e^{it}. The two opposite points on the rolling circle are \cos t and i \sin t. So, the rectangle here illustrates the fact that

e^{it} = \cos t + i \sin t

It’s interesting that this famous formula is hiding in the math of the Tusi couple! But it shouldn’t be surprising, because the Tusi couple is all about building linear motion out of circular motion… or conversely, decomposing a circular motion into linear motions.


Few of these pictures were made by me, and none of the animations. For most, you can see who created them by clicking on them. The animation of the astroid and deltoid as envelopes come from here:

Envelope, Math Images Project.

where they were made available under a GNU Free Documentation License. Here’s another animation from there:

This is a way of creating a deltoid as the envelope of some lines called ‘Wallace-Simson lines’. The definition of these lines is so baroque I didn’t dare tell it to you it earlier. But if you’re so bored you’re actually reading these credits, you might enjoy this.

Any triangle can be circumscribed by a circle. If we take any point on this circle, say M, we can drop perpendicular
lines from it to the triangle’s three sides, and get three points, say P, Q and R:

Amazingly, these points lie on a line:

This is the Wallace–Simson line of M. If we move the point M around the circle, we get lots of Wallace–Simson lines… and the envelope of these lines is a deltoid!

By the way: the Wallace–Simson line is named after William Wallace, who wrote about it, and Robert Simson, who didn’t. Don’t confuse it with the Wallace line! That was discovered by Alfred Russel Wallace.


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